- #1

SamRoss

Gold Member

- 216

- 24

## Summary:

- Is there a missing exponent in the authors' application of the Euler-Lagrange equation?

In "The Theoretical Minimum" (the one on classical mechanics), on page 218, the authors write a Lagrangian

$$L=\frac m 2 (\dot r^2 +r^2\dot \theta ^2)+\frac {GMm} r$$

They then apply the Euler-Lagrange equation ##\frac d {dt}\frac {dL} {d\dot r}=\frac {dL} {dr}## (I know there should be partial derivatives there but I couldn't find the symbol for it in my Latex primer; if anyone could enlighten me, I'd appreciate it) and wrote the result...

$$\ddot r=r\dot \theta^2-\frac {GM} r$$

My question is, shouldn't the last r in the denominator be squared since it results from differentiating the GMm/r term in the Lagrangian by r?

$$L=\frac m 2 (\dot r^2 +r^2\dot \theta ^2)+\frac {GMm} r$$

They then apply the Euler-Lagrange equation ##\frac d {dt}\frac {dL} {d\dot r}=\frac {dL} {dr}## (I know there should be partial derivatives there but I couldn't find the symbol for it in my Latex primer; if anyone could enlighten me, I'd appreciate it) and wrote the result...

$$\ddot r=r\dot \theta^2-\frac {GM} r$$

My question is, shouldn't the last r in the denominator be squared since it results from differentiating the GMm/r term in the Lagrangian by r?